A taxonomy of problems with fast parallel algorithms
Information and Control
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Capturing complexity classes by fragments of second-order logic
Theoretical Computer Science - Special issue on logic and applications to computer science
The proof theoretic strength of the Steinitz exchange theorem
Discrete Applied Mathematics
Short proofs for the determinant identities
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
| Hi-index | 0.00 |
We show that Csanky's fast parallel algorithm for computing the characteristic polynomial of a matrix can be formalized in the logical theory LAP, and can be proved correct in LAP from the principle of linear independence. LAP is a natural theory for reasoning about linear algebra introduced in [8]. Further, we show that several principles of matrix algebra, such as linear independence or the Cayley-Hamilton Theorem, can be shown equivalent in the logical theory QLA. Applying the separation between complexity classes $\textbf{AC}^0[2]\subsetneq\textbf{DET}(\text{GF}(2))$, we show that these principles are in fact not provable in QLA. In a nutshell, we show that linear independence is “all there is” to elementary linear algebra (from a proof complexity point of view), and furthermore, linear independence cannot be proved trivially (again, from a proof complexity point of view).